A

Princeton topologist and

geometer, and one of the great

expositors of

mathematics of the

20th century. Several of his short volumes of

lecture notes have become the

classic definitive expositions of their fields. These include:

Milnor and Kervaire discovered the first topological manifolds which can be given a differentiable structure (made into a smooth manifold) in several inequivalent (nondiffeomorphic) ways; they found 28 different smooth structures on the 7-sphere S^{7}. Since then many of these manifolds have been discovered. Oddly enough, every Euclidean space **R**^{n} admits a finite number of nondiffeomorphic smooth structures (the number varying with dimension), *except* **R**^{4}, which has a *continuum* of them. I don't know if there's an "explanation" *per se* of this fact, but it may be connected to the fact that 4-manifold topology and geometry seems to have a very rich theory full of hard problems, as compared to the lower-dimensional cases (which are simpler) and dimensions above 5 (where a group of *relatively* simple general principles seems to govern).